Bivariate Data: Covariance, Correlation and Contingency Tables'

1
Chapter 5

• Bivariate Data Covariance, Correlation and
  Contingency Tables.

2
Covariance and Correlation

• Notice that the two variables in our example are
  crucially related by individuals (aka
  observations)
• Covariance and correlation apply only when there
  is a set of individuals each of whom has been
  measured twice.
• It makes no sense to talk of the correlation
  between two different samples that are not
  linked together.
• Causation versus correlation

3

• The covariance is defined as the first
  cross-product moment m11 for a bivariate data set
  (x1, y1), , (xn, yn)
• Just as m2(x) is called the variance of x,
  m11(x,y) is called the covariance of x and y.
• Notice, incidentally, that m11(x,x) m2(x).

4

• The correlation coefficient is then defined as
• Notice that the factor 1/n has dropped out of rxy.

5

• Importantly, notice that

6

• Lets look at three simple examples on Eviews

• Scatterplots
• Covariance matrices
• Correlation matrices

7

• Here is how I want you to calculate the
  covariance, m11, for a given data set
• (x1, y1), , (xn, yn)
• Determine the means of x and y.
• For each i, determine (xi ) and (yi )
• For each i, determine the product pi (xi
  )(yi )
• Add up all the pi p1 p2 pn Total
• Divide by n m11 Total/n

8

• ¼(2367) ¼(18) 4.5
• ¼(0.2.4.4) ¼(1) .25
• m11 ¼(2-4.5)(0-.25) (3-4.5)(.2-.25)
  (6-4.5)(.4-.25) (7-4.5)(.4-.25)

¼(-2.5)(-.25) (-1.5)(-.05) (1.5)(.15)
(2.5)(.15) ¼(.625) (.075) (.225)
(.375) ¼1.3 .325
9

• Here is how I want you to calculate the
  correlation, r, for a given data set
• (x1, y1), , (xn, yn)
• Determine m11.
• Determine both and
• Multiply
• Divide m11/

10

• 4.5, .25, m11 .325
  ¼(0-.25)2 (.2-.25)2 (.4-.25)2
  (.4-.25)21/2
• ¼(-.25)2 (-.05)2 (.15)2 (.15)21/2

¼.0625 .0025 .0225 .02251/2
¼.111/2 .02751/2 .1658
2.06
11

• 4.5, .25, m11 .325
  .1658 2.06
• .1658(2.06) .3415

r m11/ .325/.3415 .95
12

• Then with these sorts of examples (or homeworks!)
  we plug the data into Eviews and check our
  answers.
• The correlations come out right
• Now check the standard deviations
• Whats going on?
• How do we fix this?

13

• Lets take a brief peek at some real data.

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• What is the relationship between correlation and
  covariance of x and y?
• Expressed in terms of moments, the correlation is
  the covariance divided by the product of the
  standard deviations of x and y
• Expressed in terms of standardized scores, the
  correlation between x and y is the covariance of
  the standardizations of x and y

15

• rxy is invariant over linear changes of
  measurement scales
• I.e., if x a bx, and y c dy, then rxy
  rxy.
• Also requires b, d gt 0, if we only know b, d ? 0,
  then we only have rxy rxy.
• rxy is not always invariant over nonlinear
  changes of scale
• E.g., if x x2, then (in all normal
  circumstances), rxy ? rxy
• m11 is not invariant over linear changes of
  scale
• I.e., if x a bx, and y c dy, then

16
The Correlation Coefficient

• the standardized variables are invariant to
  changes in origin and scale of the original
  variables. With standardized variables we can be
  sure that any measure of shape that we derive
  will represent the shape of the scatter plot and
  will not be an artifact of the choice of units of
  measurement. (Ramsey, p. 125 cf. p. 135)

17

• Roughly and intuitively, r measures how closely
  changes in one variable are tracked by changes in
  another variable
• More precisely, r measures the degree to which
  the y-data is a linear function of the x-data
  i.e., the degree to which yi a bxi, for some
  a and b.
• And vice-versa
• Moreover, r2 gives the exact percentage of the
  amount of the (variation in the) y-data that can
  be captured by, explained by, or projected
  onto the (variation in the) x-data.

18

• If y is a perfect function of x, i.e.,
• yi a bxi
• then rxy ?1.
• If y is not a perfect function of x, i.e., if the
  best fitting value of a and b do not capture y
  exactly
• yi a bxi ei
• then rxy lt 1.
• So while the covariance can be any number at all,
  r always lies between 1 and 1.

19
Bivariate Categorical Data
20
Terminology

• Joint probability The probability of being in
  two specific categories (one for each variable)
• e.g., being a Republican Female
• 2/12 .16

F M
R D
21
Terminology

• Marginal probability The probability of being in
  one specific category (of one of the variables)
• e.g., being a female
• 5/12 .41

F M
R D
22
Terminology

• Conditional probability The probability of being
  in one specific category (of one variable), given
  a particular value for the other variable)
• e.g., probability of a females being Republican
• 2/5 .4

F M
R D
23
Terminology

• Correlation For a 2 2 contingency table (where
  row/colum values are either 0 or 1, r reduces to
  the ? coefficient
• E.g., correlation between being female and being
  republican

F M
R D
24
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25

• What proportion of the sample is made up of
  religious persons?
• What proportion of the sample is made up of very
  happy persons?
• What proportion of the sample is made up of very
  happy religious persons?
• What proportion of very happy people are
  religious?
• What proportion of religious people are very
  happy?
• How strong is the association between being very
  happy and religious?

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• RELLIFE Please tell me whether you strongly
  agree, agree, disagree, or strongly disagree with
  the following statements I try hard to carry my
  religious beliefs over into all my other dealings
  in life.
• RELEXP Did you ever have a religious or
  spiritual experience that changed your life?
• EVOLVED Human beings, as we know them today,
  developed from earlier species of animals. (Is
  that true or false?)

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Evolved True False
Religious Experience True False
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Evolved True False
Religious Person Very/Moderate
Slight/Not
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r .42